Backbone colouring of chordal graphs

TL;DR

This paper investigates the backbone chromatic number of chordal graphs with various spanning subgraphs, establishing bounds and counterexamples for specific classes and conditions.

Contribution

It provides new bounds for the backbone chromatic number in chordal graphs with bipartite and bounded average degree subgraphs, and presents counterexamples for bipartite cases.

Findings

For interval graphs with each vertex in at most two maximal cliques, the bound holds.

Counterexamples show the bound does not extend to bipartite graphs.

Bounds are established for chordal graphs with subgraphs of bounded average degree and $C_4$-free subgraphs.

Abstract

A proper $k$-colouring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c(u)\neq c(v)$ for every edge $uv\in E(G)$. The chromatic number $\chi(G)$ is the minimum $k$ such that there exists a proper $k$-colouring of $G$. Given a spanning subgraph $H$ of $G$, a $q$-backbone $k$-colouring of $(G,H)$ is a proper $k$-colouring $c$ of $G$ such that $\lvert c(u)-c(v)\rvert \ge q$ for every edge $uv\in E(H)$. The $q$-backbone chromatic number ${\rm BBC}_q(G,H)$ is the smallest $k$ for which there exists a $q$-backbone $k$-colouring of $(G,H)$. In their seminal paper, Broersma et al.~\cite{BFGW07} ask whether, for any chordal graph $G$ and any spanning forest $H$ of $G$, we have that ${\rm BBC}_2(G,H)\leq \chi(G)+O(1)$. In this work, we first show that this is true as long as $H$ is bipartite and $G$ is an interval graph in which each vertex belongs to at most two maximal cliques. We then show that this does not extend to bipartite graphs as backbone by exhibiting a family of chordal graphs $G$ with spanning bipartite subgraphs $H$ satisfying ${\rm BBC}_2(G,H)\geq \frac{5\chi(G)}{3}$. Then, we show that if $G$ is chordal and $H$ has bounded maximum average degree (in particular, if $H$ is a forest), then ${\rm BBC}_2(G,H)\leq \chi(G)+O(\sqrt{\chi(G)})$. We finally show that ${\rm BBC}_2(G,H)\leq \frac{3}{2}\chi(G)+O(1)$ holds whenever $G$ is chordal and $H$ is $C_4$-free.